
Viewing
\((s,t)\)  as `space' and `time' coordinates in a two-dimensional Minkovski space, one may think of
the active ray \(\mathcal {R}_0(\gamma )\subset \mathcal {R}^{\rm geo}_0(\gamma )\)  as the worldline of a fictitious relativistic particle
of mass-squared \(m(\gamma )^2=r^2\Delta =\frac{1}{2} d ^2 - r \textnormal {ch}_2\) , global charge \(\gamma \)  and electric charge \(q=r\)  and immersed in a constant electric field. Indeed, when \(r\ne 0\)  the trajectory for such a particle is a branch of
hyperbola asymptoting to the light cone \(s\simeq t\operatorname{sign}r\) , while it is of course a straight line inside the light-cone when \(r=0\) . When two such particles collide, their charges add up and their mass increases by
\(m^2(\gamma +\gamma ^{\prime })-m^2(\gamma )-m^2(\gamma ^{\prime })=dd^{\prime }-r\operatorname{ch}^{\prime }_2-r^{\prime }\textnormal {ch}_2\) 
which is positive when \(\Delta ,\Delta ^{\prime }\ge 0\)  by virtue of (REF ).
This analogy is not perfect, since the initial position and velocity are
fixed by the charge (in particular, the only allowed trajectory for \(r=0\)  is a vertical line).

Two key properties following from this analogy are that the rays remain within the forward lightcone
\(|\delta s|\le \delta t\)  (the causality property), and that the `electric potential'
\(\varphi _s(\gamma ) 2(d - s r)\) 
can only increase as \(t\)  increases (for \(r=0\) , it remains constant since the ray is vertical). The fact that \(\Im Z(\gamma )=t \varphi _s(\gamma ) \)  increases with \(t\)  is an obvious consequence of the monotonicity of \(|Z(\gamma )|\)  along the attractor flow, but the point is that \(\Im Z(\gamma )\)  increases faster than \(t\) . These properties will be instrumental in the next subsection for obtaining bounds on allowed constituents for a given total charge \(\gamma \) .

In the simplest case of a collision of two incoming rays
\(\mathcal {R}^{\rm eff}_0(\gamma _1)\)  and \(\mathcal {R}^{\rm eff}_0(\gamma _2)\)  with primitive vectors \(\gamma _i\) , each of them having \(\Omega (k\gamma _i)=\delta _{k=1}\) , there is an infinite fan of outgoing effective rays of the form
\(\mathcal {R}^{\rm eff}_0(n_1\gamma _1+n_2\gamma _2)\) ,
carrying an index \(\Omega (n_1\gamma _1+n_2\gamma _2)=K_{\kappa }(n_1,n_2)\) 
given by the DT invariant of a Kronecker quiver with \(\kappa =|\langle \gamma _1,\gamma _2\rangle |\)  arrows (a multiple of 3 due to (REF )) and dimension vector \((n_1,n_2)\) 
(see Table REF  for a table of some relevant values).
More generally, the outgoing rays at each collision are determined
from the incoming rays by requiring that the product (REF ) of automorphisms
\(\mathcal {U}({\gamma _i})^{\epsilon _i}\)  around the
collision point is equal to one, or equivalently by using
the flow tree formula of [1]}, [2]},
with the indices associated to incoming rays playing the role of attractor indices.

